×
Lindskog, Filip; Resnick, Sidney I.; Roy, Joyjit

Regularly varying measures on metric spaces: hidden regular variation and hidden jumps. (English) Zbl 1317.60007

Probab. Surv. 11, 270-314 (2014).
The authors study a general framework for regular variation and the so-called heavy tails for distributions on metric-space-valued random variables, with further applications to regular variation for measures on \(\mathbb{R}_+= [0,+\infty)\) and on the space of all real-valued, right-continuous functions having left limits on \([0,1]\).
The heavy tails occur in a variety of domains such as risk management, quantitative finance and economics, complex networks of data and telecommunication transmissions, and social networks. In the particular case of one dimension, the heavy tails are known as Pareto tails. The general mathematical framework for developing the heavy tails is the theory of regular variation, originally formulated on \([0,+\infty)\) and extended to more general spaces; see [N. H. Bingham et al., Regular variation. Cambridge University Press (1987; Zbl 0617.26001); Paperback ed. (1989; Zbl 0667.26003)].
Finally, the authors attempt to clarify the proper definition of regular variation in metric spaces.
Reviewer: Sophia L. Kalpazidou (Thessaloniki)

Cited in 1 Review
Cited in 47 Documents

MSC:

60B05 Probability measures on topological spaces
28A33 Spaces of measures, convergence of measures
60G17 Sample path properties
60G51 Processes with independent increments; Lévy processes
60G70 Extreme value theory; extremal stochastic processes

Keywords:

regular variation; multivariate heavy tails; tail estimation; hidden regular variation; \(M\)-convergence; Lévy process

Citations:

Zbl 0667.26003; Zbl 0617.26001
PDF BibTeX XML Cite
\textit{F. Lindskog} et al., Probab. Surv. 11, 270--314 (2014; Zbl 1317.60007)
Full Text: DOI arXiv Euclid
  • logo of hbz
  • logo of WorldCat

References:

[1] Applebaum, D., Lévy Processes and Stochastic Calculus , volume 116 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, second edition, 2009. ISBN 978-0-521-73865-1. URL . · Zbl 1200.60001
[2] Balkema, A. A., Monotone Transformations and Limit Laws . Mathematisch Centrum, Amsterdam, 1973. Mathematical Centre Tracts, No. 45. · Zbl 0287.60023
[3] Balkema, A. A. and Embrechts, P., High Risk Scenarios and Extremes: A Geometric Approach . European Mathematical Society, 2007. · Zbl 1121.91055
[4] Barczy, M. and Pap, G., Portmanteau theorem for unbounded measures. Statistics & Probability Letters , 76(17):1831-1835, 2006. · Zbl 1109.60004
[5] Bertoin, J., Lévy Processes , volume 121 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 1996. ISBN 0-521- 56243-0.
[6] Billingsley, P., Convergence of Probability Measures . John Wiley & Sons Inc., New York, second edition, 1999. ISBN 0-471-19745-9. A Wiley-Interscience Publication. · Zbl 0944.60003
[7] Billingsley, P., Probability and Measure . Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, 2012. ISBN 978-1-118-12237-2. Anniversary edition [of MR1324786], With a foreword by Steve Lalley and a brief biography of Billingsley by Steve Koppes. · Zbl 1236.60001
[8] Bingham, N. H., Goldie, C. M., and Teugels, J. L., Regular Variation , volume 27 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, 1989. ISBN 0-521-37943-1. · Zbl 0667.26003
[9] Breiman, L., Probability , volume 7 of Classics in Applied Mathematics . Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. ISBN 0-89871-296-3. URL . Corrected reprint of the 1968 original. · Zbl 0753.60001
[10] Bruun, J. T. and Tawn, J. A., Comparison of approaches for estimating the probability of coastal flooding. J. R. Stat. Soc., Ser. C, Appl. Stat. , 47(3):405-423, 1998. · Zbl 0905.62123
[11] Coles, S. G., Heffernan, J. E., and Tawn, J. A., Dependence measures for extreme value analyses. Extremes , 2(4):339-365, 1999. · Zbl 0972.62030
[12] Daley, D. J. and Vere-Jones, D., An Introduction to the Theory of Point Processes. Vol. I . Probability and Its Applications (New York). Springer-Verlag, New York, second edition, 2003. ISBN 0-387-95541-0. Elementary theory and methods. · Zbl 1026.60061
[13] Das, B. and Resnick, S. I., Conditioning on an extreme component: Model consistency with regular variation on cones. Bernoulli , 17(1):226-252, 2011a. ISSN 1350-7265. · Zbl 1284.60103
[14] Das, B. and Resnick, S. I., Detecting a conditional extreme value model. Extremes , 14(1):29-61, 2011b. · Zbl 1226.60078
[15] Das, B., Mitra, A., and Resnick, S. I., Living on the multi-dimensional edge: Seeking hidden risks using regular variation. Advances in Applied Probability , 45(1):139-163, 2013. ArXiv e-prints 1108.5560. · Zbl 1276.60041
[16] de Haan, L., On Regular Variation and Its Application to the Weak Convergence of Sample Extremes . Mathematisch Centrum Amsterdam, 1970. · Zbl 0226.60039
[17] de Haan, L. and Ferreira, A., Extreme Value Theory: An Introduction . Springer-Verlag, New York, 2006. · Zbl 1101.62002
[18] Dudley, R. M., Real Analysis and Probability , volume 74 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2002. ISBN 0-521-00754-2. URL . Revised reprint of the 1989 original.
[19] Embrechts, P., Klüppelberg, C., and Mikosch, T., Modelling Extremal Events for Insurance and Finance . Springer-Verlag, Berlin, 2003. 4th corrected printing. · Zbl 0873.62116
[20] Geluk, J. L. and de Haan, L., Regular Variation, Extensions and Tauberian Theorems , volume 40 of CWI Tract . Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1987. ISBN 90-6196-324-9. · Zbl 0624.26003
[21] Heffernan, J. E. and Resnick, S. I., Hidden regular variation and the rank transform. Adv. Appl. Prob. , 37(2):393-414, 2005. · Zbl 1073.60057
[22] Heffernan, J. E. and Tawn, J. A., A conditional approach for multivariate extreme values (with discussion). JRSS B , 66(3):497-546, 2004. · Zbl 1046.62051
[23] Hult, H. and Lindskog, F., Extremal behavior of regularly varying stochastic processes. Stochastic Process. Appl. , 115(2):249-274, 2005. ISSN 0304-4149. · Zbl 1070.60046
[24] Hult, H. and Lindskog, F., Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd) (N.S.) , 80(94):121-140, 2006. ISSN 0350-1302. URL . · Zbl 1164.28005
[25] Hult, H. and Lindskog, F., Extremal behavior of stochastic integrals driven by regularly varying Lévy processes. Ann. Probab. , 35(1):309-339, 2007. ISSN 0091-1798. URL . · Zbl 1121.60029
[26] Hult, H., Lindskog, F., Mikosch, T., and Samorodnitsky, G., Functional large deviations for multivariate regularly varying random walks. Ann. Appl. Probab. , 15(4):2651-2680, 2005. ISSN 1050-5164. · Zbl 1166.60309
[27] Kyprianou, A. E., Fluctuations of Lévy Processes with Applications . Universitext. Springer, Heidelberg, second edition, 2014. ISBN 978-3-642-37631-3; 978-3-642-37632-0. URL . Introductory lectures. · Zbl 1384.60003
[28] Ledford, A. W. and Tawn, J. A., Statistics for near independence in multivariate extreme values. Biometrika , 83(1):169-187, 1996. ISSN 0006-3444. · Zbl 0865.62040
[29] Ledford, A. W. and Tawn, J. A., Modelling dependence within joint tail regions. J. Roy. Statist. Soc. Ser. B , 59(2):475-499, 1997. ISSN 0035-9246. · Zbl 0886.62063
[30] Maulik, K. and Resnick, S. I., Characterizations and examples of hidden regular variation. Extremes , 7(1):31-67, 2005. · Zbl 1088.62066
[31] Maulik, K., Resnick, S. I., and Rootzén, H., Asymptotic independence and a network traffic model. J. Appl. Probab. , 39(4):671-699, 2002. ISSN 0021-9002. · Zbl 1090.90017
[32] Meerschaert, M. and Scheffler, H. P., Limit Distributions for Sums of Independent Random Vectors . John Wiley & Sons Inc., New York, 2001. ISBN 0-471-35629-8. · Zbl 0077.33009
[33] Mitra, A. and Resnick, S. I., Hidden Regular Variation: Detection and Estimation. ArXiv e-prints , January 2010.
[34] Mitra, A. and Resnick, S. I., Hidden regular variation and detection of hidden risks. Stochastic Models , 27(4):591-614, 2011. · Zbl 1230.91080
[35] Mitra, A. and Resnick, S. I., Modeling multiple risks: Hidden domain of attraction. Extremes , 16:507-538, 2013. URL . · Zbl 1286.62044
[36] Peng, L., Estimation of the coefficient of tail dependence in bivariate extremes. Statist. Probab. Lett. , 43(4):399-409, 1999. ISSN 0167-7152. · Zbl 0958.62049
[37] Resnick, S. I., Point processes, regular variation and weak convergence. Adv. Applied Probability , 18:66-138, 1986. · Zbl 0597.60048
[38] Resnick, S. I., A Probability Path . Birkhäuser, Boston, 1999. · Zbl 0944.60002
[39] Resnick, S. I., Hidden regular variation, second order regular variation and asymptotic independence. Extremes , 5(4):303-336 (2003), 2002. ISSN 1386-1999. · Zbl 1035.60053
[40] Resnick, S. I., Heavy Tail Phenomena: Probabilistic and Statistical Modeling . Springer Series in Operations Research and Financial Engineering. Springer-Verlag, New York, 2007. ISBN 0-387-24272-4. · Zbl 1152.62029
[41] Resnick, S. I., Multivariate regular variation on cones: Application to extreme values, hidden regular variation and conditioned limit laws. Stochastics: An International Journal of Probability and Stochastic Processes , 80(2):269-298, 2008a. . · Zbl 1142.60042
[42] Resnick, S. I., Extreme Values, Regular Variation and Point Processes . Springer, New York, 2008b. ISBN 978-0-387-75952-4. Reprint of the 1987 original. · Zbl 1136.60004
[43] Resnick, S. I. and Zeber, D., Transition kernels and the conditional extreme value model. Extremes , 17(2):263-287, 2014. ISSN 1386-1999. URL . · Zbl 1311.60059
[44] Royden, H. L., Real Analysis . Macmillan, third edition, 1988. · Zbl 0704.26006
[45] Schlather, M., Examples for the coefficient of tail dependence and the domain of attraction of a bivariate extreme value distribution. Stat. Probab. Lett. , 53(3):325-329, 2001. · Zbl 0982.62052
[46] Seneta, E., Regularly Varying Functions . Springer-Verlag, New York, 1976. Lecture Notes in Mathematics, 508. · Zbl 0324.26002
[47] Sibuya, M., Bivariate extreme statistics. Ann. Inst. Stat. Math. , 11:195-210, 1960. · Zbl 0095.33703
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.